Plane-wave migration in tilted coordinates

Plane-wave (source plane-wave) migration Duquet et al. (2001); Liu et al. (2002); Rietveld (1995); Whitmore (1995); Zhang et al. (2005) synthesizes plane-wave source experiments from shot records. The recorded data are decomposed into plane source gathers by slant-stack processing:

$$u(x_r,z=0;p;\omega)=\int U(x_r,z=0;x_s;\omega)e^{i\omega px_s}dx_s,$$ (1)

where x_s is the source location, x_r is the receiver location, p is the ray parameter, U is the recorded surface data, and u is the synthesized surface data for the plane-wave source. The corresponding plane-source is

$$d(x_r,z=0;p;\omega)=e^{i\omega p x_s}.$$ (2)

The plane-wave source d and its corresponding synthesized data u are independently extrapolated into the subsurface, and the image can be obtained by cross-correlating these two wavefields. Plane-wave migration is potentially more efficient than shot-profile migration. It uses the whole seismic survey as the migration aperture, which is helpful for imaging steeply dipping reflectors.

 

bpvelall
Figure 2 The velocity model.

Given the plane-wave source with a ray-parameter p, its take-off angle at the surface is $\arcsin(pv)$, where v is the surface velocity. we use tilted coordinates $(x^\prime,z^\prime)$ satisfying

$$\left( \begin{array} {l} x^\prime\ z^\prime\end{array}\rig... ...array}\right) \left( \begin{array} {l} x\ z\end{array}\right),$$ (3)

where (x,z) are Cartesian coordinates and $\theta$ is an angle close to the take-off angle of the plane-wave source at the surface. In the tilted coordinates, the extrapolation direction is potentially closer to the propagation direction. Therefore, in tilted coordinates, we can extrapolate the wavefield accurately with a less accurate extrapolator. Waves that were overturned in Cartesian coordinates are not overturned in tilted coordinates. Therefore, we can image them with the one-way wave equation.